Magnets with varying magnetization direction and method of making such magnets

ABSTRACT

A permanent magnet in which the magnetization direction varies with location to optimize or restrict a magnetic field property in a selected direction at a selected point. The magnetic field property may be, for example, the transverse magnetic field, axial magnetic field, axial gradient of the transverse magnetic field, transverse gradient of the transverse magnetic field, axis gradient of the axial magnetic field, transverse gradient of the axial magnetic field, the product of the transverse magnetic field and the transverse gradient of the transverse magnetic field, the product of the transverse magnetic field and the axial gradient of the transverse magnetic field, the product of the axial magnetic field and the transverse gradient of the axial magnetic field, or the product of the axial magnetic field and the axial gradient of the axial magnetic field. The magnet may be formed of one or more segments in which the magnetization direction varies smoothly and continuously, or the magnet may be formed of a plurality of segments in which the magnetization direction is constant. A method of making and using such magnets is also disclosed.

FIELD OF THE INVENTION

This invention relates to permanent magnets, and in particular to apermanent magnet in which the magnetization direction varies to maximizea selected magnetic property of the magnet, and to methods of makingsuch magnets.

BACKGROUND OF THE INVENTION

There are a number of applications, such as magnetic surgicalapplications, where it necessary to apply strong magnetic forces (i.e.,magnetic fields and/or gradients). With recent advances in permanentmaterials, permanent magnets can provide strong magnetic forces for manyof these applications. However, the size of a conventional permanentmagnetic that is needed to provide these strong magnetic forces limitstheir usefulness. The structures needed to support massive conventionalmagnets are expensive and cumbersome. Moreover because magnetic forcesfall off rapidly (with the third power of the distance for magneticfields and with the fourth power of the distance for magneticgradients), the magnet must be very close to the point where themagnetic force is to be applied, making the rotations and translationsof the magnet to change the direction of the applied magnetic forcedifficult. In the special case of magnetic surgery applications, themovement of the magnet is also limited by imaging and life supportequipment in the procedure room.

SUMMARY OF THE INVENTION

The present invention is a permanent magnet in which the magnetizationdirection varies to maximize the selected magnetic property of themagnet (e.g. field strength or gradient), and to a method of making suchmagnets. Generally, the magnet of the present invention comprisespermanent magnets in which the magnetization direction at each point isselected to optimize the particular magnetic property. Thus in contrastto conventional permanent magnets in which the magnetization directionis uniform throughout, in the magnets of the present invention, thematerial at each point provides the optimum contribution to the desiredmagnetic property. This optimization means that a magnet of the presentinvention can be smaller and lighter than a conventional magnet, and yetsill provide equivalent magnetic force. This magnet is particularlyuseful in magnetic surgical procedures. The magnet allows smaller lesscumbersome equipment to be used in to support and manipulate the magnet,and reduces the opportunity of interference with people and equipment inthe procedure room.

According to the method of this invention, the magnet shape is firstdetermined, given the physical constraints, e.g. set off distance,accommodating other structures and equipment in the vicinity of thewhere the magnetic will be operated; and likely manipulations requiredof the magnet. A permanent magnet in the selected shape is then made, inwhich the magnetization direction at each point is generally optimizedfor the selected property. This can be accomplished by providing amonolithic permanent magnet piece with a continuously varyingmagnetization direction, or by providing a plurality of magneticsegments, which can either have a single magnetization directionthroughout, or which can themselves have continuously varyingmagnetization direction. In the former case, the magnetization directionis optimized for a particular point, for example the center of mass,thereby substantially approximating the optimum magnetization direction.

Thus magnets of the present invention provide improved magneticproperties for a given weight and size over conventional permanentmagnet. The magnets can be manipulated by smaller and less cumbersomeequipment, and due to their smaller size are less likely to interferewith, or be interfered with by persons and equipment. The method of thepresent invention provides magnets of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing the relationship between a local magneticmoment m, the magnetization angle α, and the location of the moment m asgiven by r and θ from the origin at the application point;

FIG. 2 is a top plan view of a first embodiment of a magnet, constructedaccording to the principles of this invention, optimized to provide astrong, transverse field at a selected application point spaced from themagnet face;

FIG. 3 is a top plan view of a second embodiment of a magnet constructedaccording to the principles of this invention, in which the magnet fieldwas restricted during design to provide a transverse field at a selectedapplication point spaced from the magnet face;

FIG. 4 is a top plan view of a third embodiment of a magnet, constructedaccording to the principles of this invention, optimized to provide astrong, forward field at a selected pointed spaced from the magnet face;

FIG. 5 is a top plan view of a fourth embodiment of a magnet constructedaccording to the principles of this invention, illustrating how therestriction of field direction permits asymmetry in magnet construction;

FIG. 6 is a graph of magnetization angle a versus position angle θ forrestricted b_(x) (i.e., b_(y)=0) (the magnet shown in FIG. 3), and forthe optimized of b_(x) (the magnet shown in FIG. 2);

FIG. 7 is a graph of the scaled magnetic field component in the xdirection (b′_(x)) versus position angle θ of the element for arestricted field magnetization case (b_(y)=0), the scaled magnet fieldcomponent in the y direction (b′_(y)) being zero;

FIG. 8 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the field in the x direction at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the restricted fieldmagnetization curve (b_(y)=0));

FIG. 9 is a graph of magnetization angle α versus position angle θ forthe restricted gradient (∂b_(x)/∂x=0) and for the restricted gradient(∂b_(x)/∂y=0);

FIG. 10 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ of the element for the restrictedtransverse field gradient in the axial direction case (∂b_(x)/∂y=0);

FIG. 11 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ of the element for the restrictedtransverse field gradient in the transverse direction case(∂b_(x)/∂y=0);

FIG. 12 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the axial gradient component of the transverse field (i.e.,the field in the x direction) at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the restricted gradientmagnetization case (∂b_(x)/∂x=0);

FIG. 13 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the transverse gradient component of the transverse field(i.e., the field in the x direction) at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the restricted gradientmagnetization case (∂b_(x)/∂y=0);

FIG. 14 is a graph of the scaled magnetic field component in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ of the element for the optimized fieldmagnetization case (b_(x) maximized)(e.g. for the magnet shown in FIG.2);

FIG. 15 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the field in the x direction at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the optimized fieldmagnetization case (b_(x) maximized);

FIG. 16 is graph of magnetization angle α versus position angle θ foroptimized gradient component (∂b_(x)/∂x optimized with respect to α),and for optimized gradient component (∂b_(x)/∂y optimized with respectto α);

FIG. 17 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ of the element for the optimizedtransverse gradient of the transverse field (∂b_(x)/∂x optimized withrespect to α);

FIG. 18 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) versus position angleθ for the optimized axial gradient of the transverse field (∂b_(x)/∂yoptimized with respect to α);

FIG. 19 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the optimized transverse gradient of the transverse field(i.e., the field in the x direction) at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the optimized gradientmagnetization case (∂b_(x)/∂x optimized);

FIG. 20 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the optimized axial gradient of the transverse field (i.e.,the field in the x direction) at the origin, (i.e., an angulardistribution curve r vs. θ as a polar plot) for the optimized gradientmagnetization case (∂b_(x)/∂y optimized);

FIG. 21 is graph of magnetization angle α versus position angle θ foroptimized field-gradient product components (b_(x)(∂b_(x)/∂x) optimized)and (b_(x)(∂b_(x)/∂y) optimized);

FIG. 22 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ for the element for optimizedfield-gradient product magnetization case (b_(x)(∂b_(x)/∂x) optimized);

FIG. 23 is a graph of the scaled magnetic field components in the xdirection (b′_(x)) and in the y direction (b′_(y)) of the contributingelement versus position angle θ for optimized field-gradient productmagnetization case (b_(x)(∂b_(x)/∂y) maximized);

FIG. 24 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the optimized field gradient product at the origin, (i.e., anangular distribution curve r vs. θ as a polar plot) for the optimizedgradient magnetization case (b_(x)(∂b_(x)/∂x) optimized);

FIG. 25 is a constant contribution curve, i.e., the curve representingthe positions where a properly aligned magnetic moment would contributeequally to the optimized field gradient product at the origin, (i.e., anangular distribution curve r vs. θ as a polar plot) for the optimizedgradient magnetization case [b_(x)(∂b_(x)/∂y) maximized];

FIG. 26A is a perspective view of a magnet constructed from a pluralityof permanent magnetic segments, according to the principles of thisinvention;

FIG. 26B is top plan view of the magnet shown in FIG. 26A;

FIG. 26C is a side elevation view of the magnet shown in FIG. 26A;

FIG. 26D is a front elevation view of the magnet shown in FIG. 26A;

FIG. 27A is a front perspective view of an alternate construction of amagnet constructed from a plurality of permanent magnet segments,according to the principles of this invention; and

FIG. 27B is a rear perspective view of the magnet shown in FIG. 27A.

Corresponding reference numerals indicate corresponding parts throughoutthe several views of the drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The magnets of the preferred embodiments of the present invention arepermanent magnets in which the magnetization direction varies bylocation to provide magnets that optimize a selected magnetic fieldproperty at a selected location external to the magnet. These magnetsmay comprise a monolithic magnet in which the magnetization directionvaries smoothly and continuously so that at each location, themagnetization direction is substantially in the direction that optimizesthe selected magnetic property. Alternatively, these magnets maycomprise a plurality of magnet segments, each with a magnetization thatis substantially in the direction that optimizes the selected magneticfield property. These magnets can be used in any application, but areparticularly useful for magnetic surgical applications.

The methods of making magnets of the preferred embodiments of thepresent invention provide for the simple construction of optimizedmagnets. The methods involve the arrangement of individual magneticdipole elements. These dipole elements may be actual microscopicmagnetic domains in a material, or they may be macroscopic magneticsegments assembled to form the magnet. Using the dipole approximation,it is possible to derive formulas for the magnetization direction as afunction of the position. Based upon these formulas the appropriatemagnetization direction at any location can be determined. Thismagnetization can then be achieved by variably magnetizing a monolithicmagnet, or by assembling a plurality of magnetic segments of differentuniform magnetization directions.

A magnetic dipole is approximated by the formula: $\begin{matrix}{b = {\frac{\mu_{o}}{4\quad\pi}\left\lbrack {{- \frac{m}{r^{3}}} + \frac{3\left( {m \cdot r} \right)r}{r^{5}}} \right\rbrack}} & (1)\end{matrix}$where b is the magnetic field due to the elemental magnetic moment m, ris the location of the moment, and μ_(o) is the permeability of freespace. The orientation of m for each location in the magnet isdetermined to maximize a selected magnetic field property at theselected operating point. The collection of N magnetic momentscomprising the magnet create a source field B, that is the sum of thesource fields created by each of the magnetic moments: $\begin{matrix}{B = {\sum\limits_{i = 1}^{N}\quad b_{i}}} & (2)\end{matrix}$

In this preferred embodiment, the collection of magnetic moments thatform the magnet are treated in two dimensions, i.e., in a plane. Theoptimization of the magnet moments which represent the localmagnetization direction, will be two-dimensional, and the importantelemental magnetizations will therefore lie in this chosen plane. Whilea full three-dimensional optimization has been carried out, thetwo-dimensional case is described as the preferred embodiment becausemagnets with constant magnetization in one direction are simpler toconstruct, and three-dimensional optimization yields only an incrementalgain in efficiency. It is apparent to one of ordinary skill in the artthat the optimization could be a three-dimensional, which although insome cases might be important, in most cases does not yield importantincreases in efficiency.

As discussed herein, the coordinates are chosen so that the active planein which the moment orientations are restricted is designated the x-yplane. Then, in principle, the z-axis contains no variation inmagnetization within the magnet boundaries, although it can beappreciated that boundary conditions may render this assumption slightlyinaccurate depending on the restrictions or parameter optimization.Therefore, the optimal magnetization directions are easily analyzed inthe plane defined by the magnetic field b and the magnetic moment m.

FIG. 1 shows the coordinate system in which the microscopic moment m, isat a magnetization angle α, and a location given by r and θ. Thedirection of the vector r is reversed from its usual textbook usage.This is permissible since r appears quadratically in the momentequation. It is useful in the present application, since it leadsintuitively to the usage of its two-dimensional magnitude as an angulardistribution of magnetized moments needed to provide required fieldsand/or gradients. Using the coordinate system in FIG. 1, b of equation(1) can be expressed in terms of the angles αand θ. $\begin{matrix}{b = {\begin{Bmatrix}b_{x} \\b_{y}\end{Bmatrix} = {\frac{\mu_{o}m}{8\quad\pi\quad r^{3}}\begin{Bmatrix}{{A\quad\cos\quad\alpha} + {B\quad\sin\quad\alpha}} \\{{B\quad\cos\quad\alpha} + {\left( {2 - A} \right)\sin\quad\alpha}}\end{Bmatrix}}}} & (3) \\{{\partial b_{x}} = {\begin{Bmatrix}{{\partial b_{x}}/{\partial x}} \\{{\partial b_{x}}/{\partial y}}\end{Bmatrix} = {\frac{3\quad\mu_{o}m}{8\quad\pi\quad r^{4}}\begin{Bmatrix}{{C\quad\cos\quad\alpha} + {D\quad\sin\quad\alpha}} \\{{E\quad\cos\quad\alpha} + {F\quad\sin\quad\alpha}}\end{Bmatrix}}}} & (4) \\{where} & \quad \\\left. \begin{matrix}{{A \equiv {A(\theta)}} = {1 + {3\quad\cos\quad 2\quad\theta}}} \\{{B \equiv {B(\theta)}} = {3\quad\sin\quad 2\quad\theta}}\end{matrix} \right\} & (5) \\\left. \begin{matrix}{{C \equiv {C(\theta)}} = {{2\quad\sin\quad\theta\quad\sin\quad 2\quad\theta} - {\cos\quad\theta} - {3\quad\cos\quad\theta\quad\cos\quad 2\quad\theta}}} \\{{D \equiv {D(\theta)}} = {{3\quad\cos\quad\theta\quad\sin\quad 2\quad\theta} + {2\quad\sin\quad\theta\quad\cos\quad 2\quad\theta}}} \\{{E \equiv {E(\theta)}} = {{{- 2}\quad\cos\quad\theta\quad\sin\quad 2\quad\theta} - {\sin\quad\theta} - {3\quad\sin\quad\theta\quad\cos\quad 2\quad\theta}}} \\{{F \equiv {F(\theta)}} = {{2\quad\cos\quad\theta\quad\cos\quad 2\quad\theta} - {3\quad\sin\quad\theta\quad\sin\quad 2\quad\theta}}}\end{matrix} \right\} & (6)\end{matrix}$Thus the quantities A, B, C, D, E and F are functions only of theelement position angle θ relative to a coordinate system in which theelemental field b and complete field B are described. Furthermore, thequantities given by equations (3) and (4) represent the complete reducedset given that ∇·b=0 and ∇×b=0, thereby establishing that there are onlyfour unique quantities b_(x), the field in the x direction, b_(y), thefield in the y direction, ∂b_(x)/∂x, the gradient in the x direction,and ∂b_(x)/∂y, the gradient in the y direction, (assuming, as statedabove, that the z components are ignored).

Co-pending U.S. patent application Ser. No. 09/497,467, filed Feb. 3,2000, entitled An Efficient Magnet System for Magnetically-AssistedSurgery, incorporated herein by reference, details one embodiment inwhich the completed magnet is operated so that the procedure point in apatient is on the central x-y plane.

The essence of the methods described herein is to select the desiredmagnetic field properties of the completed magnet, and apply therequisite implied conditions to the individual elemental moments. Forexample, if a strong, flat, transverse, central field region is neededfor the operating region of a magnet, the individual elemental moments(at angle α) can be aligned relative to their locations (i.e., asspecified by angle θ in the plane) so as to produce optimalcontributions to that type of field. This is a unique vector type of“focusing” to accomplish a desired field projection for each element ofthe magnet, adding up to an optimum for the complete magnet. Asdescribed in more detail below, the first step in development of amagnet is to calculate the relationship between α and θ for the selectedmagnetic field property and selected location, for the magnet. In thecase of a predominantly transverse field over the operating centerregion of the magnet, one requisite implied condition would be to setb_(y) equal to zero in equation (3) above. FIGS. 2 and 4 showalternative magnet constructions for optimizing the magnetic field alongthe x-direction. In FIG. 2 the field is parallel to the magnet face. InFIG. 4, the field is perpendicular to the magnet face. In the case ofthe restricted field magnets (shown in FIGS. 3 and 5) the magnetsprovide strong fields in a given direction by setting perpendicularcomponents to zero at the outset. In the case of optimized field magnets(shown in FIGS. 2 and 4) the magnets provide the mathematical optimumfiled strength. The restricted field magnets, automatically achieves thedesired field direction, irrespective of magnet shape or symmetry (asillustrated by FIG. 5). The optimal field magnet provides the optimalfield strength in the desired direction, for a given size, but requiressymmetric retention of elements to preserve field direction.

In another application, the requirement may be for a gradient that iszero except in a selected direction in a selected region spaced from themagnet. The magnetic field gradient is a tensor, so in two-dimensions,two such cases can exist mathematically, for ∂b_(x)/∂x=0 or for∂b_(x)/∂y=0, by the use of equation (4) above. (∂b_(y)/∂x=∂b_(x)/∂y from∇×b=0.) Clearly, a number of magnet shapes could be used with thisrestricted case.

In still another application, the requirement may be for theoptimization of the magnetic field gradients in some region. And stillanother application, the requirement may be for the field-gradientproducts to be optimized. This last case would be useful for the pullingof permeable materials or objects in some particular direction, whileorienting them in the same or a different direction. Five majorcategories of these restrictions and optimizations are discussed below.One of ordinary skill in the art could use similar methods for othercases.

Once the optimal magnet shape and the magnetization direction for agiven position are determined, a permanent magnet can be fabricated froma permanent magnetic material, such as a neodymium-boron-iron (Nd—B—Fe)material, of the appropriate shape can be magnetized in the appropriatedirections, or a magnet can be assembled from a plurality if magneticsegments, with the proper shape and magnetization direction.

Specifically, the five major categories of cases employ two differentprocedures: restriction and optimization, although similar results cansometimes be achieved by either procedure. The two procedures differ,organizationally, in the use either of restriction of an undesiredproperty, or the optimization of a desired property. The details ofelement selection at the end to achieve a final complete magnet will bemotivated somewhat differently in the two procedures.

Cases A and B first restrict components of certain magnetic fieldproperties so as to a priori provide the directions needed for thedesired properties. The case will then, at the end, use material removalto make the magnet as small and light as possible while achievingrequired strengths of the desired quantities at the desired location.

Alternatively, cases C, D and E will first optimize (maximize) thedesired quantities, and at the end remove material to simultaneouslymake the magnet smaller and to rid the field of undesired“off-direction” components of the desired quantity, since the procedurewill not have automatically removed them. Clearly, attention to symmetryis important.

A. Restricted Field Case

This case first restricts a magnetic field property, uses equations toarrive at a prototype magnet shape, and then uses judicious materialremoval to optimize the desired quantity in conjunction with achievingthe smallest and lightest magnet, typically using an iterative or trialand error method. This case does not use a formal optimizationprocedure.

In this case, one component of the magnetic field of the element iszeroed at a distance from the magnet. This restricted case is useful inthat the magnet may be composed of segments that do not need to bebalanced (i.e., symmetry need not be reflected in the magnet's materialremoval, or design) to ensure that one component of the field remainszeroed.

The procedure here will be first to develop equations which relate themagnetization angle α to the position angle θ for a particular elementso that it will contribute a magnetic field only in the x direction,i.e. so b_(y)=0, for some point of interest in the operating region.This can be thought of as a conditional requirement on each particularelement, to be used later in the assembly of elements for the magnet.This might, for example, be a case where a transverse field was neededin the patient procedure volume. FIGS. 6, 7, and 8 will be used todepict, respectively and as described above, the consequences of thisrestriction on 1) the magnetization angle-position angle relationship,2) the relative magnetic field component contributions vs. positionangle θ at an implicit but correctly scaled distance r, and 3) a polarplot of the distance r at which the given reference field strength isreached in the x-y plane.

Setting b_(y)=0, we find that $\begin{matrix}{{\tan\quad\alpha} = \frac{B}{A - 2}} & (7)\end{matrix}$relates the magnetization angle α to the element position angle θ(implicitly through A and B). This is depicted as the solid line in FIG.3. From symmetry, the case b_(x)=0 can be seen by extrapolation. FIG. 6contains the information which is needed to keep track of (in thecomputer) the magnetization directions α in the “shape” plot or “angulardistribution” plot, FIG. 8 below, which is used to determine the finalmagnet shape. FIG. 7 portrays the change in the magnetic fieldcontributions of the elements due to their varying magnetizationdirections (as a consequence of varying θ) where the plotted field isscaled so that$b^{\prime} = {\left( \frac{8\quad\pi\quad r^{3}}{\mu_{o}m} \right){b.}}$It also follows that for constant contributions of b_(x), the desiredshape of the magnet is given by solving the x-component of equation (3)for r: $\begin{matrix}{r = {\left( \frac{\mu_{o}m}{8\quad\pi\quad b_{x}} \right)^{1/3}\left( {{A\quad\cos\quad\alpha} + {B\quad\sin\quad\alpha}} \right)^{1/3}}} & (8)\end{matrix}$for which θ, again, is implicit in A and B. This is shown by the polarplot of FIG. 5 (i.e., r versus θ) where the radial component is scaledby the quantity (8πb_(x)/μ_(o)m)^(1/3). A few of the directions α(implicit in r(θ)) are shown by the arrows around the r(θ) curve. Themagnet shown in FIG. 3 is derived from this restricted field case.

In summary, the magnet shape and ideal magnetization distribution isdetermined in the x-y plane from FIG. 8, which represents a locus of theplanar value of r at which a reference value of b_(x) is achieved.

B. Restricted Gradient Case

Similar in overall method to the restricted field case, this case, inaddition to practically restricting a different quantity, is useful inthat the magnet may then be composed of segments that do not requiresymmetry in their design to remove the influence of undesirable gradientterms at the point r. For the choice of gradient direction null a∂b_(x)/∂x=0, we find that $\begin{matrix}{{\tan\quad\alpha} = {- \frac{C}{D}}} & (9)\end{matrix}$and for ∂b_(x)/∂y=0, it follows that $\begin{matrix}{{\tan\quad\alpha} = {- \frac{E}{F}}} & (10)\end{matrix}$FIG. 9 depicts the results of equations (9) and (10), showing α versus θexplicitly in the two sub-cases. FIGS. 10 and 11 plot the relationshipof the scaled magnetic field contributions to θ for the two cases∂b_(x)/∂x=0 and ∂b_(x)/∂y=0, respectively. In FIG. 9 there are now twoα-θ relationships, corresponding to the two directions of gradientrestriction. It is notable, and relevant to final magnet design, thatwhile θ was double-valued in a for the restricted field condition, it isnow triple-valued in α in each of the gradients not restricted to zero.The two curves in FIG. 10 for the scaled field component (b_(x)′ andb_(y)′) contributions, both of which now survive, oscillate with θ butare 90 degrees out of phase. In FIG. 11 the scaled magnetic fieldcomponents contributions again oscillate out of phase, but with only asingle cycle in 360 degrees of θ.

Given a constant gradient component ∂b_(x)/∂y, polar plots for theradial component r for which ∂b_(x)/∂x=0 are shown in FIG. 12 where$\begin{matrix}{r = {\left\lbrack \frac{3\quad\mu_{o}m}{8\quad{\pi\left( {{\partial b_{x}}/{\partial y}} \right)}} \right\rbrack^{1/4}\left( {{E\quad\cos\quad\alpha} + {F\quad\sin\quad\alpha}} \right)^{1/4}}} & (11)\end{matrix}$Likewise, a constant ∂b_(x)/∂x for which ∂b_(x)/∂y=0 yields FIG. 13where now $\begin{matrix}{r = {\left\lbrack \frac{3\quad\mu_{o}m}{8\quad{\pi\left( {{\partial b_{x}}/{\partial x}} \right)}} \right\rbrack^{1/4}\left( {{C\quad\cos\quad\alpha} + {D\quad\sin\quad\alpha}} \right)^{1/4}}} & (12)\end{matrix}$In these restricted cases, the correct 1/r⁴ dependence of the gradientis retained so that the appropriate magnitudes are preserved. Themulti-lobed shape curves (angular distribution curves) require more carein the use of symmetries, etc. in the design of the complete magnet,e.g. the choice of where to remove material.C. Maximized Field Cases

Instead of restricting a field or gradient component at some selectedfocal point (operating region), it may be more useful to use analternative, formal optimization method to achieve a similar but betterresult. Here the distributions are calculated for element orientationswhich are formally optimized to maximize a field component. For example,it will optimize (maximize) a field component to replace the a priorirestricting of an undesired component as was the case in case A. Theresulting optimized magnet shapes can be seen by comparing the magnet inFIG. 2 (optimized) and FIG. 3 (restricted). In this new case, we wish torelate α to θ for the magnetic moments to be oriented in such a manneras to contribute maximally to either b_(x) or b_(y), depending on thedesired direction of the field relative to the magnet face. Given thatthe field components are analogous if θ sweeps from 0 to 2π, ∂b_(x)/∂α=0yields: $\begin{matrix}{{\tan\quad\alpha} = {\frac{B}{A}.}} & (13)\end{matrix}$The relationship between α and θ for this optimized field case is shownin dashed lines in FIG. 6. FIG. 14 details the relationship of themagnetic field component contributions to θ. Obviously, care must betaken in the design of the complete magnet by judicious choice ofsymmetry to ensure the removal of the y-component of the field which nowis not automatically excluded at the start. The nature of r is identicalin form to that expressed in equation (8). However, since therelationship between α and θ is now given by equation (13) rather thanby equation (7), we see that r versus θ now takes the simple form ofFIG. 15, which is different from the form resulting from the restrictedfield case shown in FIG. 8. It should also be noted that for a maximumto exist, ∂²b_(x)/∂α²<0 which results in the requirement that b_(x)>0,an obvious condition.

Thus, the new result yields the magnet shape shown in FIG. 2, as opposedto the shape shown in FIG. 3 for the restricted field case, to arrive ata particular field strength for a strictly transverse field parallel tothe magnet face at some given distance. It is apparent that the newmethod yields a smaller optimal magnet cross section, and a lightermagnet in this particular type of design problem.

D. Optimized Gradient Case

Here the overall procedure parallels that of case C. But now thegradient components, not the field components, are formally maximized,and the material removal is used (with symmetry considerations) at theend to achieve the desired gradient directions.

As in the procedures above, we relate α to θ so that the moment of theelement is oriented in a manner to contribute maximally to the requiredquantity, except that here that quantity is either ∂b_(x)/∂x or∂b_(x)/∂y, For${\frac{\partial\quad}{\partial\alpha}\left( \frac{\partial b_{x}}{\partial x} \right)} = 0$we calculate $\begin{matrix}{{\tan\quad\alpha} = \frac{D}{C}} & (14)\end{matrix}$which yields the α-θ relationship shown in solid lines in FIG. 16. Themagnetic field component contribution is depicted in FIG. 17 and thevariable r takes the form of equation (12), but with the new implicitα-θ relationship changing the result. That is plotted in FIG. 19.$\begin{matrix}{{{{For}\quad\frac{\partial\quad}{\partial\alpha}\left( \frac{\partial b_{x}}{\partial y} \right)} = {0\text{:}}}{{\tan\quad\alpha} = \frac{F}{E}}} & (15)\end{matrix}$which yields the α-θ relationship shown in dashed lines in FIG. 16. NowFIG. 18 plots the magnetic fields and the behavior of r is in the formof equation (11) and depicted in FIG. 20. As discussed above, theignored gradients and field components must be examined in determiningless important material removal in the magnet's design, depending on theapplication of the magnet.E. Maximized Field-Gradient Product Case

Again, as in cases C and D, the overall procedure is to formallymaximize a desired quantity and finally to use symmetry conditions onmaterial removal to achieve the desired magnetic field property ofinterest.

It is often of interest to provide a pulling force on a permeablematerial or medium. A good quantity in most cases for evaluating such aforce is the field-gradient product. Therefore in this final major case,it is desired to maximize the quantities$b_{x}\frac{\partial b_{x}}{\partial y}\quad{and}{\quad\quad}b_{x}\frac{\partial b_{x}}{\partial x}$which represent the only unique components of the field-gradientproduct. All other cases for the field-gradient product follow from∇·b=0 and ∇×b=0. From equations (3) and (4) we rewrite thefield-gradient product in the form of $\begin{matrix}{{b_{x}\frac{\partial b_{x}}{\partial y}} = {\frac{3}{2r^{7}}\left( \frac{\mu_{o}m}{8\quad\pi} \right)^{2}{\quad\left\lbrack {\left( {{AE} + {BF}} \right) + {\left( {{AE} - {BF}} \right)\cos\quad 2\quad\alpha} + {\left( {{BE} + {AF}} \right)\sin\quad 2\quad\alpha}} \right\rbrack}}} & (16) \\{{b_{x}\frac{\partial b_{x}}{\partial x}} = {\frac{3}{2r^{7}}\left( \frac{\mu_{o}m}{8\quad\pi} \right)^{2}{\quad\left\lbrack {\left( {{AC} + {BD}} \right) + {\left( {{AC} - {BD}} \right)\cos\quad 2\quad\alpha} + {\left( {{BC} + {AD}} \right)\sin\quad 2\quad\alpha}} \right\rbrack}}} & (17)\end{matrix}$Thus for${{\frac{\partial\quad}{\partial\alpha}\left( {b_{x}\frac{\partial b_{x}}{\partial y}} \right)} = 0},$we find that $\begin{matrix}{{\tan\quad 2\quad\alpha} = \frac{{BE} + {AF}}{{AE} - {BF}}} & (18)\end{matrix}$and for${{\frac{\partial\quad}{\partial\alpha}\left( {b_{x}\frac{\partial b_{x}}{\partial x}} \right)} = 0},$the relationship between α and θ becomes $\begin{matrix}{{\tan\quad 2\quad\alpha} = \frac{{BC} + {AD}}{{AC} - {BD}}} & (19)\end{matrix}$where the condition for a maximum to exists for both equations (16) and(17) is that tan² 2α≧−1. The relationship between α and θ, fromequations (18) and (19), is shown in FIG. 21 in solid liens forb_(x)db_(x)/dy and in dashed lines for b_(x)db_(x/)d_(x), and the fielddistribution plots for the two sub-cases are in FIGS. 22 (b_(x)db_(x)/dy) and 23 (b_(x)db_(x)/dx). The two-dimensional r(θ) curves detailed inequations (20) and (21) follow from (16) and (17), respectively:$\begin{matrix}\left. {r = {{{\left\lbrack {\frac{3}{2{b_{x}\left( {{\partial b_{x}}/{\partial y}} \right)}}\left( \frac{\mu_{0}m}{8\quad\pi} \right)^{2}} \right\rbrack^{1/7}\left\lbrack {\left( {{AE} + {BF}} \right) + {AE} - {BF}} \right)}\quad\cos\quad 2\quad\alpha} + {\left( {{BE} + {AF}} \right)\quad\sin\quad 2\quad\alpha}}} \right\rbrack^{1/7} & (20) \\{r = {\left\lbrack {\frac{3}{2{b_{x}\left( {{\partial b_{x}}/{\partial x}} \right)}}\left( \frac{\mu_{0}m}{8\quad\pi} \right)^{2}} \right\rbrack^{1/7}\left\lbrack {\left( {{AC} + {BD}} \right) + {\left( {{AC} - {BD}} \right)\quad\cos\quad 2\quad\alpha} + {\left( {{BC} + {AD}} \right)\quad\sin\quad 2\quad\alpha}} \right\rbrack}^{1/7}} & (21)\end{matrix}$These are shown as polar plots in FIGS. 24 and 25, respectively. Caremust be paid to the symmetry of the magnetic field components andgradients in the magnet's design if pure field-gradient terms are to begenerated (e.g., for b_(x)(∂b_(x)/∂y) maximized with b_(y)=0 and∂b_(x)/∂x=0).Operation In designing a magnet in accordance with the principles ofthis invention, one would first generally select the desired shape ofthe magnet. This is most conveniently done with two dimensionaloptimization by using the curves of constant contribution for themagnetic field property that is to be optimized or restricted, to givethe shape in two dimensions, i.e., in the x-y plane. The height in thez-axis can then be selected based on the available space, desiredweight, and required field strength. Once the shape of the magnet isgenerally determined, the magnetization direction at each location isthen determined. For optimization in two dimensions, the relationsbetween the magnetization direction angle α verses the position angle θ,discussed above can be use to determine the proper magnetization angleat each location.

In the case of a monolithic magnetic, a blank of permanent magneticmaterial can be made in the desired shape and magnetized so that themagnetization direction varies smoothly and continuously in the x-yplane in accordance with the proper magnetization direction determinedby the appropriate relation between the magnetization direction angle αverses the position angle. As noted above, in most cases themagnetization need not vary in three dimensions, and thus the directionof magnetizations vary only in the x-y plane. This means that the magnetcan be made of a single monolithic block or a plurality of similarpermanent magnet slabs stacked in the non-critical (i.e., the zdirection).

In the case of a multi-segmented magnet, a plurality of permanent magnetsegments are assembled to conform substantially to the desired shape.The magnetization direction of each segment is selected so that themagnetization direction within the segment conforms substantially to theproper magnetization direction. Ideally the magnetization would varywithin the segment to conform to the proper magnetization direction thatsatisfied the optimization/restriction criteria for the desired magneticfield property. However, achieving the ideal is technically difficultand expensive, and with judicious selection of the sizes and shapes ofthe permanent magnet segments makes this unnecessary. A single uniformmagnetization direction can usually be selected for each segment thatsatisfactorily approximates the ideal magnetization directiondistribution. For example, the magnetization direction for a segmentcould be selected to be the ideal direction at the location of thecenter of mass of the segment. The magnetization direction for thesegment could alternatively be selected to be the ideal direction at thelocation of the magnetic center of the segment, (i.e. the location of atwhich a single magnet dipole of equivalent strength could replace thesegment), or the material in the segment could be otherwise weighted(for example, inversely to the cube of the distance between each pointin the segment and the selected point), to provide the appropriate pointfor in the segment to determine the proper magnetization direction toapply to the entire segment.

The monolithic magnets in accordance with this invention ideally wouldhave cross-sectional shapes in the x-y plane corresponding to the shapeof the constant contribution for the particular magnetic field propertybeing optimized or restricted. This provides the most efficient magnetfor the volume and weight. Thus, at least some of the surfaces of themagnets of this invention, and in particular the surfaces that face awayfrom the selected point of application of the optimized/restrictedmagnetic field property, conform to the surface of constantcontribution. However, as a practical matter it can be difficult to formmagnetic material into this shapes, and in many cases where volume andweight are not critical, it is not necessary that the magnet have thisshape. As long as the magnetization direction is properly orientedthroughout the magnet, the additional material does not impair theoptimized/restricted magnetic field property, it merely makes the magnetlarger and heavier than necessary.

Where size and/or weight of the magnet is critical, then the constantcontribution curves provide guidance of where to remove material withthe least impact on the magnet field properties. More specifically for agiven constant contribution surface, it is always more efficient toremove material outside the surface before removing material inside thesurface, even if the material outside the surface is closer to theselected point than other material inside the surface. Thus, forconvenience magnets can be made in square or rectangular blocks thateither approximate the optimal curved surfaces, or that are machined toapproximate the optimal curved surfaces.

For example, in making a magnet that maximizes the transverse field, anappropriate curve of constant contribution for the magnet size andstrength is used. See FIG. 8. The setoff distance between the magnet andthe selected operating point is selected, and the cross section of themagnet in the x-y plane is then determined. The properties of the magnetfield that would be created by such a magnet can be calculated todetermine if the magnet is the appropriate size. If the properties arelarger than required, a smaller magnet can be selected. If theproperties are smaller than required, a larger magnet can be selected.The magnet can then be completed by providing a monolithic block orpermanent magnet material of the selected cross section, or a pluralityof slabs of the selected cross-section, and magnetizing the block or theslabs in the direction α in the x-y plane determined by the positionangle θ, given by relation between α and θ for the desired optimizationor restriction of the magnetic field properties. Of course, rather thanusing a monolithic block, or a plurality of slabs stacked in the zdirection, the magnets can be assembled from a plurality of segments.The segments can be shaped to form the magnetic of the desired shape(see FIGS. 26A-D), or where the size and weight are not important, theycan simply be rectangular prisms, which as easier to work with (seeFIGS. 27A and B). Each segment has the appropriate magnetizationdirection α in the x-y plane for the location of the segment θ. Ofcourse, for ease of manufacture, the magnet can be made with rectangularsegments as shown in FIG. 27, and machined or otherwise shaped to theoptimal shape, such as that shown in FIG. 26.

1. A permanent magnet in which the magnetization direction varies inthree dimensions with location to optimize a desired magnetic fieldproperty in a selected direction at a selected point.
 2. The permanentmagnet in accordance with claim 1 wherein the desired magnetic fieldproperty is selected from transverse magnetic field, axial magneticfield, axial gradient of the transverse magnetic field, transversegradient of the transverse magnetic field, axis gradient of the axialmagnetic field, transverse gradient of the axial magnetic field, theproduct of the transverse magnetic field and the transverse gradient ofthe transverse magnetic field, the product of the transverse magneticfield and the axial gradient of the transverse magnetic field, theproduct of the axial magnetic field and the transverse gradient of theaxial magnetic field, or the product of the axial magnetic field and theaxial gradient of the axial magnetic field.
 3. A permanent magnet inwhich the magnetization direction varies in three dimensions to restrictan undesired magnetic field property in a selected direction at aselected point
 4. The permanent magnet in accordance with claim 3wherein the undesired magnetic field property is selected fromtransverse magnetic field, axial magnetic field, axial gradient of thetransverse magnetic field, transverse gradient of the transversemagnetic field, axis gradient of the axial magnetic field, transversegradient of the axial magnetic field, the product of the transversemagnetic field and the transverse gradient of the transverse magneticfield, the product of the transverse magnetic field and the axialgradient of the transverse magnetic field, the product of the axialmagnetic field and the transverse gradient of the axial magnetic field,or the product of the axial magnetic field and the axial gradient of theaxial magnetic field.
 5. A permanent magnet in which the magnetizationdirection varies in three dimensions with location to optimize a desiredmagnetic field property in a selected direction at a selected point. 6.The permanent magnet in accordance with claim 5 wherein the desiredmagnetic field property is selected from transverse magnetic field,axial magnetic field, axial gradient of the transverse magnetic field,transverse gradient of the transverse magnetic field, axis gradient ofthe axial magnetic field, transverse gradient of the axial magneticfield, the product of the transverse magnetic field and the transversegradient of the transverse magnetic field, the product of the transversemagnetic field and the axial gradient of the transverse magnetic field,the product of the axial magnetic field and the transverse gradient ofthe axial magnetic field, or the product of the axial magnetic field andthe axial gradient of the axial magnetic field.
 7. A permanent magnet inwhich the magnetization direction varies in three dimensions to restrictan undesired magnetic field property in a selected direction at aselected point.
 8. The permanent magnet in accordance with claim 3wherein the undesired magnetic field property is selected fromtransverse magnetic field, axial magnetic field, axial gradient of thetransverse magnetic field, transverse gradient of the transversemagnetic field, axis gradient of the axial magnetic field, transversegradient of the axial magnetic field, the product of the transversemagnetic field and the transverse gradient of the transverse magneticfield, the product of the transverse magnetic field and the axialgradient of the transverse magnetic field, the product of the axialmagnetic field and the transverse gradient of the axial magnetic field,or the product of the axial magnetic field and the axial gradient of theaxial magnetic field.
 9. A method of performing a medical procedureusing the magnet of claim 1 to project magnetic field into a patient tocontrol a magnetic medical element inside the patient.
 10. A method ofperforming a medical procedure using the magnet of claim 3 to project amagnetic field into a patient to control a magnetic medical elementinside the patient.
 11. A method of making a permanent magnet in whichthe magnetization direction varies with location to optimize a desiredmagnetic field property at a selected point in a selected direction, themethod comprising: determining the desired shape, providing a blank ofpermanent magnetic material in the desired shape, and magnetizing themagnet to have a magnetization direction that varies in three dimensionsso that the magnetization at each location in the magnet is in thedirection that substantially optimizes the desired magnetic fieldproperty at a selected point in the selected direction.
 12. A method ofmaking a permanent magnet in which the magnetization direction varieswith location to restrict an undesired magnetic field property at aselected point in a selected direction, the method comprising:determining the desired shape, providing a blank of permanent magneticmaterial in the desired shape, and magnetizing the magnet to have amagnetization direction that varies in three dimensions so that themagnetization at each location in the magnet is in the direction thatsubstantially restricts the undesired magnetic field property at aselected point in the selected direction.
 13. A method of making apermanent magnet in which the magnetization direction varies withlocation to optimize a desired magnetic field property at a selectedpoint in a selected direction, the method comprising: determining thedesired shape, providing a blank of permanent magnetic material in thedesired shape, and magnetizing the magnet to have a magnetizationdirection that varies in two dimensions so that the magnetization ateach location in the magnet is in the direction that substantiallyoptimizes the desired magnetic field property at a selected point in theselected direction.
 14. A method of making a permanent magnet in whichthe magnetization direction varies with location to restrict anundesired magnetic field property at a selected point in a selecteddirection, the method comprising: determining the desired shape,providing a blank of permanent magnetic material in the desired shape,and magnetizing the magnet to have a magnetization direction that variesin two dimensions so that the magnetization at each location in themagnet is in the direction that substantially restricts the undesiredmagnetic field property at a selected point in the selected direction.15. A method of making a permanent magnet in which the magnetizationdirection varies with location to optimize a desired magnetic fieldproperty at a selected point in a selected direction, the methodcomprising: determining the desired shape, assembling a plurality ofpermanent magnet segments into a shape substantially conforming to thedesired shape, the magnetization direction of each permanent magnetsegment varying in three dimensions so that the magnetization directionof each permanent magnet segment is in the direction that substantiallyoptimizes the desired magnetic field property at a selected point in theselected direction.
 16. A method of making a permanent magnet in whichthe magnetization direction varies with location to restrict anundesired magnetic field property at a selected point in a selecteddirection, the method comprising determining the desired shape,assembling a plurality of permanent magnet segments into a shapesubstantially conforming to the desired shape, the magnetizationdirection of each permanent magnet segment varying in three dimensionsso that the magnetization direction of each permanent magnet segment isin the direction that substantially restricts the undesired magneticfield property at a selected point in the selected direction.
 17. Amethod of making a permanent magnet in which the magnetization directionvaries with location to optimize a desired magnetic field property at aselected point in a selected direction, the method comprising:determining the desired shape, assembling a plurality of permanentmagnet segments into a shape substantially conforming to the desiredshape, the magnetization direction of each permanent magnet segmentvarying in two dimensions so that the magnetization direction of eachpermanent magnet segment is in the direction that substantiallyoptimizes the desired magnetic field property at a selected point in theselected direction.
 18. A method of making a permanent magnet in whichthe magnetization direction varies with location to restrict anundesired magnetic field property at a selected point in a selecteddirection, the method comprising determining the desired shape,assembling a plurality of permanent magnet segments into a shapesubstantially conforming to the desired shape, the magnetizationdirection of each permanent magnet segment varying in two dimensions sothat the magnetization direction of each permanent magnet segment is inthe direction that substantially restricts the undesired magnetic fieldproperty at a selected point in the selected direction.
 19. The methodaccording to claim 18 wherein at least a portion of the surface of themagnet conforms to a surface of constant contribution to the desiredmagnetic field property at the selected location.
 20. The methodaccording to claim 18 wherein the direction of magnetization throughouteach permanent magnet segment is constant.
 21. The method according toclaim 20 wherein the direction of magnetization throughout eachpermanent magnet segment is the direction which, at the center of massof the segment, provides the maximum contribution to the desiredproperty.
 22. The method according to claim 20 wherein the direction ofmagnetization throughout each permanent magnet segment is the directionwhich, at the effective magnet center, provides the maximum contributionto the desired property.
 23. The method according to claim 20 whereinthe size and position of the permanent magnet segments is selected sothat the difference in the direction of magnetization direction betweenadjacent magnet segments is less than about 45°.
 24. The methodaccording to claim 23 wherein the size and position of the permanentmagnet segments is selected so that the difference in the direction ofmagnetization direction between adjacent magnet segments is less thanabout 30°.
 25. The method according to claim 20 wherein themagnetization direction throughout each permanent magnet segment is notconstant.
 26. The method according to claim 20 wherein the desiredmagnetic field property is selected from transverse magnetic field,axial magnetic field, axial gradient of the transverse magnetic field,transverse gradient of the transverse magnetic field, axis gradient ofthe axial magnetic field, transverse gradient of the axial magneticfield, the product of the transverse magnetic field and the transversegradient of the transverse magnetic field, the product of the transversemagnetic field and the axial gradient of the transverse magnetic field,the product of the axial magnetic field and the transverse gradient ofthe axial magnetic field, or the product of the axial magnetic field andthe axial gradient of the axial magnetic field.
 27. The method accordingto claim 20 wherein the desired property is transverse magnetic field istransverse magnetic field, and the relationship between themagnetization and angle the angular position is given by equation (3).28. A method of making a permanent magnet in which the magnetizationdirection varies to control a desired property of the magnetic field ata selected point; determining the desired magnetization direction α as afunction of θ the angle from the selected point to control the desiredproperty of the magnetic field; selecting a cross section for the magnetin a plane containing the selected direction providing a magneticmaterial in substantially the selected cross section, in which themagnetization direction α, substantially conforms to the function ateach θ.
 29. The method according to claim 28 wherein the magneticmaterial is monolithic with a continuously variable magnetizationdirection.
 30. The method according to claim 29 wherein the magneticmaterial comprises a plurality of discrete magnet segments, with themagnetization direction of each segment having a constant magnetizationdirection that substantially conforms to magnetization direction α as afunction of θ through the magnet segment.